Now we pass to the description of the second order analysis overP2(Rd). The concepts that now enter into play are: Covariant Derivative, Parallel Transport and Curvature. To some extent, the situation is similar to the one we discussed in Subsection 2.3.2 concerning the first order structure: the metric space(P2(Rd), W2)is not a Riemannian manifold, but if we are careful in giving definitions and in the regularity requirements of the objects involved we will be able to perform calculations very similar to those valid in a genuine Riemannian context.
Again, we are restricting the analysis to the Euclidean case only for simplicity: all of what comes next can be generalized to the analysis overP2(M), for a generic Riemannian manifoldM.
On a typical course of basic Riemannian geometry, one of the first concepts introduced is that of Levi-Civita connection, which identifies the only natural (“natural” here means: “compatible with the Riemannian structure”) way of differentiating vector fields on the manifold. It would therefore be natural to set up our discussion on the second order analysis onP2(Rd)by giving the definition of Levi-Civita connection in this setting. However, this cannot be done. The reason is that we don’t have a notion of smoothness for vector fields, therefore not only we don’t know how to covariantly differentiate vector fields, but we don’t know either which are the vector fields regular enough to be differentiated. In a purely Riemannian setting this problem does not appear, as a Riemannian man- ifold borns as smooth manifold on which we define a scalar product on each tangent space; but the spaceP2(Rd)does not have a smooth structure (there is no diffeomorphism of a small ball around the origin inTanµ(P2(Rd))onto a neighborhood ofµinP2(Rd)). Thus, we have to proceed in a different way, which we describe now:
Regular curvesfirst of all, we drop the idea of defining a smooth vector field on the whole “mani- fold”. We will rather concentrate on finding an appropriate definition of smoothness for vector fields defined along curves. We will see that to do this, we will need to work with a particular kind of curves, which we callregular, see Definition 6.2.
Smoothness of vector fields. We will then be able to define the smoothness of vector fields defined along regular curves (Definition 6.5). Among others, a notion of smoothness of particular relevance is that ofabsolutely continuousvector fields: for this kind of vector fields we have a natural notion oftotal derivative(not to be confused with the covariant one, see Definition 6.6).
Levi-Civita connection. At this point we have all the ingredients we need to define the covariant derivative and to prove that it is the Levi-Civita connection onP2(Rd)(Definiton 6.8 and discussion thereafter).
Parallel transport. This is the main existence result on this subject: we prove that along regular curves the parallel transport always exists (Theorem 6.15). We will also discuss a counterexample to the existence of parallel transport along a non-regular geodesic (Example 6.16). This will show that the definition of regular curve is not just operationally needed to provide a definition of smoothness
of vector fields, but is actually intrinsically related to the geometry ofP2(Rd).
Calculus of derivatives. Using the technical tools developed for the study of the parallel transport, we will be able to explicitly compute the total and covariant derivatives of basic examples of vector fields.
Curvature. We conclude the discussion by showing how the concepts developed can lead to a rigor- ous definition of the curvature tensor onP2(Rd).
We will writekvkµ andhv, wiµfor the norm of the vector fieldvand the scalar product of the vector fieldsv, win the spaceL2(µ)(which we will denote byL2µ), respectively.
We now start with the definition of regular curve. All the curves we will consider are defined on [0,1], unless otherwise stated.
Definition 6.2 (Regular curve) Let(µt)be an absolutely continuous curve and let(vt)be its ve- locity vector field, that is (vt) is the unique vector field - up to equality for a.e. t - such that vt∈Tanµt(P2(Rd))for a.e.tand the continuity equation
d
dtµt+∇ ·(vtµt) = 0,
holds in the sense of distributions (recall Theorem 2.29 and Definition 2.31). We say that (µt)is regular provided
Z 1 0
kvtk2µtdt <∞, (6.2)
and
Z 1 0
Lip(vt)dt <∞. (6.3)
Observe that the validity of (6.3) is independent on the parametrization of the curve, thus if it is fulfilled it is always possible to reparametrize the curve (e.g. with constant speed) in order to let it satisfy also (6.2).
Now assume that(µt)is regular. Then by the classical Cauchy-Lipschitz theory we know that there exists a unique family of mapsT(t, s,·) : supp(µt)→supp(µs)satisfying
( d
dsT(t, s, x) = vs(T(t, s, x)), ∀t∈[0,1], x∈supp(µt), a.e. s∈[0,1], T(t, t, x) = x, ∀t∈[0,1], x∈supp(µt).
(6.4) Also it is possible to check that these maps satisfy the additional properties
T(r, s,·)◦T(t, r,·) = T(t, s,·) ∀t, r, s∈[0,1], T(t, s,·)#µt = µs, ∀t, s∈[0,1].
We will call this family of maps theflow mapsof the curve(µt). Observe that for any couple of timest, s∈[0,1], the right composition withT(t, s,·)provides a bijective isometry fromL2µ
sto L2µ
t. Also, notice that from condition (6.2) and the inequalities kT(t, s,·)−T(t, s0,·)k2µt ≤
Z Z s0 s
vr(T(t, r, x))dr
!2 dµt(x)
≤ |s0−s|
Z s0 s
kvr(x)k2µr(x)dr
we get that for fixedt∈[0,1], the maps7→T(t, s,·)∈L2µ
tis absolutely continuous.
It can be proved that the set of regular curves is dense in the set of absolutely continuous curves onP2(Rd)with respect to uniform convergence plus convergence of length. We omit the technical proof of this fact and focus instead on the important case of geodesics:
Proposition 6.3 (Regular geodesics) Let(µt)be a constant speed geodesic on [0,1]. Then its re- striction to any interval[ε,1−ε], withε >0, is regular. In general, however, the whole curve(µt) may be not regular on[0,1].
Proof To prove that(µt)may be not regular just consider the case ofµ0 :=δxandµ1:= 12(δy1+ δy2): it is immediate to verify that for the velocity vector field(vt)it holdsLip(vt) =t−1.
For the other part, recall from Remark 2.25 (see also Proposition 2.16) that fort ∈ (0,1)and s∈[0,1]there exists a unique optimal mapTtsfromµttoµs. It is immediate to verify from formula (2.11) that these maps satisfy
Tts−Id
s−t =Tts0−Id
s0−t , ∀t∈(0,1), s∈[0,1].
Thus, thanks to Proposition 2.32, we have thatvtis given by vt= lim
s→t
Tts−Id
s−t =Id−Tt0
t . (6.5)
Now recall that Remark 2.25 givesLip(T0t)≤(1−t)−1to obtain Lip(vt)≤t−1((1−t)−1+ 1) = 2−t
t(1−t).
Thust7→Lip(vt)is integrable on any interval of the kind[ε,1−ε],ε >0.
Definition 6.4 (Vector fields along a curve) A vector field along a curve (µt) is a Borel map (t, x)7→ut(x)such thatut∈L2µ
tfor a.e.t. It will be denoted by(ut).
Observe that we are considering also non tangent vector fields, that is, we are not requiring ut∈Tanµt(P2(Rd))for a.e.t.
To define the (time) smoothness of a vector field(ut)defined along a regular curve(µt)we will make an essential use of the flow maps: notice that the main problem in considering the smoothness of(ut)is that for different times, the vectors belong to different spaces. To overcome this obstruction we will define the smoothness oft7→ut∈L2µt in terms of the smoothness oft7→ut◦T(t0, t,·)∈ L2µt
0:
Definition 6.5 (Smoothness of vector fields) Let(µt)be a regular curve, T(t, s,·) its flow maps and(ut)a vector field defined along it. We say that(ut)is absolutely continuous (orC1, orCn,. . ., orC∞or analytic) provided the map
t7→ut◦T(t0, t,·)∈L2µt
0
is absolutely continuous (orC1, orCn,. . ., orC∞or analytic) for everyt0∈[0,1].
Sinceut◦T(t1, t,·) =ut◦T(t0, t,·)◦T(t1, t0,·)and the composition withT(t1, t0,·)provides an isometry fromL2µ
t0 toL2µ
t1, it is sufficient to check the regularity oft7→ut◦T(t0, t,·)forsome t0∈[0,1]to be sure that the same regularity holds for everyt0.
Definition 6.6 (Total derivative) With the same notation as above, assume that(ut)is an absolutely continuous vector field. Its total derivative is defined as:
d
dtut:= lim
h→0
ut+h◦T(t, t+h,·)−ut
h ,
where the limit is intended inL2µ
t.
Observe that we are not requiring the vector field to be tangent, and that the total derivative is in general a non tangent vector field, even if(ut)is.
The identity lim
h→0
ut+h◦T(t, t+h,·)−ut
h =
lim
h→0
ut+h◦T(t0, t+h,·)−ut◦T(t0, t,·) h
◦T(t, t0,·)
= d
dt ut◦T(t0, t,·)
◦T(t, t0,·),
shows that the total derivative is well defined for a.e.tand that is anL1vector field, in the sense that it holds
Z 1 0
d dtut
µ
t
dt <∞.
Notice also the inequality
kus◦T(t, s,·)−utkµt ≤ Z s
t
d
dt(ur◦T(t, r,·)) µ
t
dr= Z s
t
d dtur
µ
r
dr.
An important property of the total derivative is theLeibnitz rule: for any couple of absolutely contin- uous vector fields(u1t),(u2t)along the same regular curve(µt)the mapt7→
u1t, u2t
µtis absolutely continuous and it holds
d dt
u1t, u2t
µt = d
dtu1t, u2t
µt
+
u1t, d dtu2t
µt
, a.e. t. (6.6)
Indeed, from the identity u1t, u2t
µt =
u1t◦T(t0, t,·), u2t◦T(t0, t,·)
µt0, it follows the absolute continuity, and the same expression gives
d dt
u1t, u2t
µt = d dt
u1t◦T(t0, t,·), u2t◦T(t0, t,·)
µt0
= d
dt u1t◦T(t0, t,·)
, u2t◦T(t0, t,·)
µt0
+
u1t◦T(t0, t,·), d
dt u2t◦T(t0, t,·)
µt0
= d
dtu1t, u2t
µt
+
u1t, d dtu2t
µt
.
Example 6.7 (The smooth case) Let (x, t) 7→ ξt(x)be a Cc∞ vector field on Rd,(µt) a regular curve and(vt)its velocity vector field. Then the inequality
kξs◦T(t, s,·)−ξtkµt ≤ kξs−ξtkµs+kξt◦T(t, s,·)−ξtkµt≤C|s−t|+C0kT(t, s,·)−Idkµt,
withC:= supt,x|∂tξt(x)|,C0:= supt,x|ξt(x)|, together with the fact thats7→T(t, s,·)∈L2(µt) is absolutely continuous, gives that(ξt)is absolutely continuous along(µt).
Then a direct application of the definition gives that its total derivative is given by d
dtξt=∂tξt+∇ξt·vt, a.e. t, (6.7) which shows that the total derivative is nothing but the convective derivativewell known in fluid
dynamics.
Forµ∈P2(Rd), we denote byPµ :L2µ →Tanµ(P2(Rd))the orthogonal projection, and we putP⊥µ :=Id−Pµ.
Definition 6.8 (Covariant derivative) Let(ut)be an absolutely continuous andtangentvector field along the regular curve(µt). Its covariant derivative is defined as
D
dtut:= Pµt d
dtut
. (6.8)
The trivial inequality
D dtut
µ
t
≤
d dtut
µ
t
shows that the covariant derivative is anL1vector field.
In order to prove that the covariant derivative we just defined is the Levi-Civita connection, we need to prove two facts: compatibiliy with the metricandtorsion free identity. Recall that on a standard Riemannian manifold, these two conditions are respectively given by:
d
dthX(γt), Y(γt)i=
(∇γt0X)(γt), Y(γt) +
X(γt),(∇γt0Y)(γt) [X, Y] =∇XY − ∇YX,
whereX, Y are smooth vector fields andγis a smooth curve onM.
The compatibility with the metric follows immediately from the Leibnitz rule (6.6), indeed if (u1t),(u2t)are tangent absolutely continuous vector fields we have:
d dt
u1t, u2t
µt = d
dtu1t, u2t
µt
+
u1t, d dtu2t
µt
=
Pµt
d dtu1t
, u2t
µt
+
u1t,Pµt
d dtu2t
µt
= D
dtu1t, u2t
µt
+
u1t,D dtu2t
µt
.
(6.9)
To prove the torsion-free identity, we need first to understand how to calculate the Lie bracket of two vector fields. To this aim, letµit,i= 1,2, be two regular curves such thatµ10=µ20=:µand let uit∈Tanµi
t(P2(Rd))be twoC1vector fields satisfyingu10=v20,u20=v10, wherevtiare the velocity vector fields ofµit. We assume that the velocity fieldsvitofµitare continuous in time (in the sense that the mapt7→vtiµitis continuous in the set of vector valued measure with the weak topology and t 7→ kvtikµi
t is continuous as well), to be sure that (6.7) holds foralltwithvt = vitand the initial condition makes sense. With these hypotheses, it makes sense to consider the covariant derivative
D
dtu2talong(µ2t)att= 0: for this derivative we write∇u1
0u2t. Similarly for(u1t).
Let us consider vector fields as derivations, and the functionalµ7→Fϕ(µ) := R
ϕdµ, for given ϕ ∈ Cc∞(Rd). By the continuity equation, the derivative ofFϕalongu2t is equal to
∇ϕ, u2t
µ2t, therefore the compatibility with the metric (6.9) gives:
u1(u2(Fϕ))(µ) = d dt
∇ϕ, u2t
µ2t|t=0=
∇2ϕ·v20, u20
µ+D
∇ϕ,∇u1 0u2tE
µ
=
∇2ϕ·u10, u20
µ+D
∇ϕ,∇u1 0u2tE
µ.
Subtracting the analogous termu2(u1(Fϕ))(µ)and using the symmetry of∇2ϕwe get [u1, u2](Fϕ)(µ) =D
∇ϕ,∇u1
0u2t−∇u2
0u1tE
µ
.
Given that the set{∇ϕ}ϕ∈C∞c is dense inTanµ(P2(Rd)), the above equation characterizes[u1, u2] as:
[u1, u2] =∇u1
0u2t−∇u2
0u1t, (6.10)
which proves the torsion-free identity for the covariant derivative.
Example 6.9 (The velocity vector field of a geodesic) Let (µt) be the restriction to [0,1] of a geodesic defined in some larger interval(−ε,1 +ε)and let(vt)be its velocity vector field. Then we know by Proposition 6.3 that(µt)is regular. Also, from formula (6.5) it is easy to see that it holds
vs◦T(t, s,·) =vt, ∀t, s∈[0,1],
and thus(vt)is absolutely continuous and satisfiesdtdvt= 0and a fortioriDdtvt= 0.
Thus, as expected, the velocity vector field of a geodesic has zero convariant derivative, in analogy with the standard Riemannian case. Actually, it is interesting to observe that not only the covariant
derivative is 0 in this case, but also the total one.
Now we pass to the question of parallel transport. The definition comes naturally:
Definition 6.10 (Parallel transport) Let(µt)be a regular curve. A tangent vector field(ut)along it is a parallel transport if it is absolutely continuous and
D
dtut= 0, a.e. t.
It is immediate to verify that the scalar product of two parallel transports is preserved in time, indeed the compatibility with the metric (6.9) yields
d dt
u1t, u2t
µt = D
dtu1t, u2t
µt
+
u1t,D dtu2t
µt
= 0, a.e. t,
for any couple of parallel transports. In particular, this fact and the linearity of the notion of parallel transport give uniqueness of the parallel transport itself, in the sense that for any u0 ∈ Tanµ0(P2(Rd))there exists at most one parallel transport(ut)along(µt)satisfyingu0=u0.
Thus the problem is to show the existence. There is an important analogy, which helps under- standing the proof, that we want to point out: we already know that the space(P2(Rd), W2)looks like a Riemannian manifold, but actually it has also stronger similarities with a Riemannian manifold M embedded in some bigger space (say, on some Euclidean spaceRD), indeed in both cases:
• we have a natural presence of non tangent vectors: elements of L2µ \ Tanµ(P2(Rd))for P2(Rd), and vectors inRDnon tangent to the manifold for the embedded case.
• The scalar product in the tangent space can be naturally defined also for non tangent vectors:
scalar product inL2µ for the spaceP2(Rd), and the scalar product inRD for the embedded case. This means in particular that there are natural orthogonal projections from the set of tangent and non tangent vectors onto the set of tangent vectors: Pµ : L2µ →Tanµ(P2(Rd)) forP2(Rd)andPx:RD→TxM for the embedded case.
• The Covariant derivative of a tangent vector field is given by projecting the “time derivative”
onto the tangent space. Indeed, for the spaceP2(Rd)we know that the covariant derivative is given by formula (6.8), while for the embedded manifold it holds:
∇γ˙tut=Pγt
d dtut
, (6.11)
wheret7→γtis a smooth curve andt7→ut∈TγtM is a smooth tangent vector field.
Given these analogies, we are going to proceed as follows: first we give a proof of the existence of the parallel transport along a smooth curve in an embedded Riemannian manifold, then we will see how this proof can be adapted to the Wasserstein case: this approach should help highlighting what’s the geometric idea behind the construction.
Thus, say thatM is a given smooth Riemannian manifold embedded onRD, t 7→ γt ∈ M a smooth curve on[0,1]andu0∈Tγ0M is a given tangent vector. Our goal is to prove the existence of an absolutely continuous vector fieldt7→ut∈TγtM such thatu0=u0and
Pγt d
dtut
= 0, a.e. t.
For anyt, s ∈ [0,1], lettrst : TγtRD → TγsRD be the natural translation map which takes a vector with base pointγt(tangent or not to the manifold) and gives back the translated of this vector with base pointγs. Notice that an effect of the curvature of the manifold and the chosen embedding onRD, is thattrst(u)may be not tangent toM even ifuis. Now definePts:TγtRD→TγsM by
Pts(u) :=Pγs(trst(u)), ∀u∈TγtRD.
An immediate consequence of the smoothness ofM andγare the two inequalities:
|trst(u)−Pts(u)| ≤C|u||s−t|, ∀t, s∈[0,1]andu∈TγtM, (6.12a)
|Pts(u)| ≤C|u||s−t|, ∀t, s∈[0,1]andu∈Tγ⊥tM, (6.12b) whereTγ⊥tM is the orthogonal complement ofTγtM inTγtRD. These two inequalities are all we need to prove existence of the parallel transport. The proof will be constructive, and is based on the identity:
∇γtP0t(u)|t=0= 0, ∀u∈Tγ(0)M, (6.13) which tells that the vectorsP0t(u)are a first order approximation att = 0of the parallel transport.
Taking (6.11) into account, (6.13) is equivalent to
|Pt0(trt0(u)−P0t(u))|=o(t), u∈Tγ(0)M. (6.14) Equation (6.14) follows by applying inequalities (6.12) (note thattrt0(u)−P0t(u)∈Tγ⊥tM):
|Pt0(trt0(u)−P0t(u))| ≤Ct|trt0(u)−P0t(u)| ≤C2t2|u|.
Now, letPbe the direct set of all the partitions of[0,1], where, forP,Q ∈ P,P ≥ QifP is a refinement ofQ. ForP ={0 =t0< t1<· · ·< tN = 1} ∈Pandu∈Tγ0MdefineP(u)∈Tγ1M as:
P(u) :=PttN−1N (PttN−2N−1(· · ·(P0t1(u)))).
Our first goal is to prove that the limitP(u)forP ∈ Pexists. This will naturally define a curve t→ut∈TγtMby taking partitions of[0, t]instead of[0,1]: the final goal is to show that this curve is actually the parallel transport ofualong the curveγ.
The proof is based on the following lemma.
Lemma 6.11 Let0≤s1≤s2≤s3≤1be given numbers. Then it holds:
Pss13(u)−Pss23(Pss12(u))
≤C2|u||s1−s2||s2−s3|, ∀u∈Tγs1M.
Proof FromPss3
1(u) =Pγs
3(trss3
1(u)) =Pγs
3(trss3
2(trss2
1(u)))we get Pss3
1(u)−Pss3
2(Pss2
1(u)) =Pss3
2(trss2
1(u)−Pss2
1(u)) Sinceu∈Tγs1M andtrss21(u)−Pss12(u)∈Tγ⊥s
2M, the proof follows applying inequalities (6.12).
From this lemma, an easy induction shows that for any0≤s1<· · ·< sN ≤1andu∈Tγs1M we have
PssN
1 (u)−PssN−1N (PssN−2N−1(· · ·(Pss2
1(u))))
≤
Pss1N(u)−PssN−1N (Pss1N−1(u)) +
PssN−1N (Pss1N−1(u))−PssN−1N (PssN−2N−1(· · ·(Pss12(u))))
≤ C2|u||sN1−s1|!sN −sN−1|+
Pss1N−1(u)−PssN−2N−1(· · ·(Pss12(u)))
≤ · · ·
≤ C2|u|
N−1
X
i=2
|s1−si||si−si+1| ≤C2|u||s1−sN|2. (6.15) With this result, we can prove existence of the limit ofP(u)asP varies inP.
Theorem 6.12 For anyu∈Tγ0M there exists the limit ofP(u)asPvaries inP.
Proof We have to prove that, givenε >0, there exists a partitionP such that
|P(u)− Q(u)| ≤ |u|ε, ∀Q ≥ P. (6.16) In order to do so, it is sufficient to find0 =t0< t1<· · ·< tN = 1such thatP
i|ti+1−ti|2≤ε/C2, and repeatedly apply equation (6.15) to all partitions induced byQin the intervals(ti, ti+1).
Now, fors ≤twe can introduce the mapsTts : TγtM →TγsM which associate to the vector u ∈ TγtM the limit of the process just described taking into account partitions of[s, t]instead of those of[0,1].
Theorem 6.13 For anyt1≤t2≤t3∈[0,1]it holds
Ttt23◦Ttt12 =Ttt13. (6.17) Moreover, for anyu ∈ Tγ0M the curvet → ut := T0t(u) ∈ TγtM is the parallel transport ofu alongγ.
Proof For the group property, consider those partitions of[t1, t3]which containt2 and pass to the limit first on[t1, t2]and then on[t2, t3]. To prove the second part of the statement, we prove first that(ut)is absolutely continuous. To see this, pass to the limit in (6.15) withs1=t0andsN =t1, u=ut0to get
|Ptt01(ut0)−ut1| ≤C2|ut0|(t1−t0)2≤C2|u|(t1−t0)2, (6.18) so that from (6.12a) we get
|trtt10(ut0)−ut1| ≤ |trtt10(ut0)−Ptt01(ut0)|+|Ptt01(ut0)−ut1| ≤C|u||t1−t0|(1 +C|t1−t0|), which shows the absolute continuity. Finally, due to (6.17), it is sufficient to check that the covariant derivative vanishes at 0. To see this, putt0= 0andt1=tin (6.18) to get|P0t(u)−ut| ≤C2|u|t2,
so that the thesis follows from (6.13).
Now we come back to the Wasserstein case. To follow the analogy with the Riemannian case, keep in mind that the analogous of the translation maptrst is the right composition withT(s, t,·), and the analogous of the mapPtsis
Pts(u) := Pµs(u◦T(s, t,·)), which mapsL2µ
t ontoTanµs(P2(Rd))We saw that the key to prove the existence of the parallel transport in the embedded Riemannian case are inequalities (6.12). Thus, given that we want to im- itate the approach in the Wasserstein setting, we need to produce an analogous of those inequalities.
This is the content of the following lemma.
We will denote byTan⊥µ(P2(Rd))the orthogonal complement ofTanµ(P2(Rd))inL2µ. Lemma 6.14 (Control of the angles between tangent spaces) Letµ, ν ∈P2(Rd)andT :Rd → Rdbe any Borel map satisfyingT#µ=ν. Then it holds:
kv◦T−Pµ(v◦T)kµ≤ kvkνLip(T−Id), ∀v∈Tanν(P2(Rd)), and, ifT is invertible, it also holds
kPµ(w◦T)kµ≤ kwkνLip(T−1−Id), ∀w∈Tan⊥ν(P2(Rd)).
Proof We start with the first inequality, which is equivalent to
k∇ϕ◦T−Pµ(∇ϕ◦T)kµ≤ k∇ϕkνLip(T−Id), ∀ϕ∈Cc∞(Rd). (6.19) Let us suppose first that T−Id ∈ Cc∞(Rd). In this case the mapϕ◦T is inCc∞(Rd), too, and therefore∇(ϕ◦T) =∇T·(∇ϕ)◦T belongs toTanµ(P2(Rd)). From the minimality properties of the projection we get:
k∇ϕ◦T−Pµ(∇ϕ◦T)kµ≤ k∇ϕ◦T− ∇T·(∇ϕ)◦Tkµ
= Z
|(I− ∇T(x))· ∇ϕ(T(x))|2dµ(x) 1/2
≤ Z
|∇ϕ(T(x))|2k∇(Id−T)(x)k2opdµ(x) 1/2
≤ k∇ϕkνLip(T−Id),
whereIis the identity matrix andk∇(Id−T)(x)kopis the operator norm of the linear functional fromRdtoRdgiven byv7→ ∇(Id−T)(x)·v.
Now turn to the general case, and we can certainly assume thatT is Lipschitz. Then, it is not hard to see that there exists a sequence(Tn−Id) ⊂ Cc∞(Rd)such that Tn → T uniformly on compact sets andlimnLip(Tn −Id)≤ Lip(T −Id). It is clear that for such a sequence it holds kT−Tnkµ→0, and we have
k∇ϕ◦T−Pµ(∇ϕ◦T)kµ≤ k∇ϕ◦T− ∇(ϕ◦Tn)kµ
≤ k∇ϕ◦T− ∇ϕ◦Tnkµ+k∇ϕ◦Tn− ∇(ϕ◦Tn)kµ
≤Lip(∇ϕ)kT−Tnkµ+k∇ϕ◦TnkµLip(Tn−Id).
Lettingn→+∞we get the thesis.
For the second inequality, just notice that kPµ(w◦T)kµ= sup
v∈Tanµ(P2 (Rd)) kvkµ=1
hw◦T, viµ= sup
v∈Tanµ(P2 (Rd)) kvkµ=1
w, v◦T−1
ν
= sup
v∈Tanµ(P2 (Rd)) kvkµ=1
w, v◦T−1−Pν(v◦T−1)
ν ≤ kwkνLip(T−1−Id) From this lemma and the inequality
Lip
T(s, t,·)−Id
≤e|RtsLip(vr)dr| −1≤C
Z s t
Lip(vr)dr
, ∀t, s∈[0,1], (whose simple proof we omit), whereC:=eR01Lip(vr)dr−1, it is immediate to verify that it holds:
ku◦T(s, t,·)−Pts(u)kµs ≤Ckukµt
Z s t
Lip(vr)dr
, u∈Tanµt(P2(Rd)), kPts(u)kµs ≤Ckukµt
Z s t
Lip(vr)dr
, u∈Tan⊥µt(P2(Rd)).
(6.20)
These inequalities are perfectly analogous to the (6.12) (well, the only difference is that here the bound on the angle is L1 int, swhile for the embedded case it wasL∞, but this does not really change anything). Therefore the arguments presented before apply also to this case, and we can derive the existence of the parallel transport along regular curves:
Theorem 6.15 (Parallel transport along regular curves) Let (µt) be a regular curve and u0 ∈ Tanµ0(P2(Rd)). Then there exists a parallel transport(ut)along(µt)such thatu0=u0.
Now, we know that the parallel transport exists along regular curves, and we know also that regular curves are dense, it is therefore natural to try to construct the parallel transport along any absolutely continuous curve via some limiting argument. However, this cannot be done, as the fol- lowing counterexample shows:
Example 6.16 (Non existence of parallel transport along a non regular geodesic) Let
Q = [0,1]×[0,1]be the unit square in R2 and letTi, i = 1,2,3,4, be the four open trian- gles in whichQis divided by its diagonals. Letµ0 :=χQL2and define the functionv:Q→R2 as the gradient of the convex mapmax{|x|,|y|}, as in the figure. Set alsow=v⊥, the rotation by π/2ofv, inQandw= 0out ofQ. Notice that∇ ·(wµ0) = 0.
Setµt := (Id+tv)#µ0 and observe that, for positive t, the support Qtof µt is made of 4 connected components, each one the translation of one of the setsTi, and thatµt=χQtL2.
It is immediate to check that(µt)is a geodesic in [0,∞), so that from 6.3 we know that the restriction ofµtto any interval[ε,1]withε >0is regular. Fixε >0and note that, by construction, the flow maps ofµtin[ε,1]are given by
T(t, s,·) = (Id+sv)◦(Id+tv)−1, ∀t, s∈[ε,1].
Now, set wt := w◦T(t,0,·) and notice thatwt is tangent atµt (because wtis constant in the connected components of the support of µt, so we can define aCc∞function to be affine on each connected component and with gradient given bywt, and then use the space between the components themselves to rearrange smoothly the function). Sincewt+h◦T(t, t+h,·) =wt, we havedtdwt= 0 and a fortiori Ddtwt= 0. Thus(wt)is a parallel transport in[ε,1]. Furthermore, since∇ ·(wµ0) = 0, we havew0=w /∈Tanµ0(P2(R2)). Therefore there is no way to extendwtto a continuoustangent vector field on the whole[0,1]. In particular, there is no way to extend the parallel transport up to
t= 0.
Now we pass to the calculus of total and covariant derivatives. Let(µt)be a fixed regular curve and let(vt)be its velocity vector field. Start observing that, if(ut)is absolutely continuous along